Disassociated theories of correspondences: fractured cognitive connectivity

Theories of Correspondences -- and potential equivalences between them in correlative thinking (Part #2)

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The challenge of comprehending the connectivity of "moonshine" is well-illustrated by use of any web search for "theory of correspondences". Two contrasting sets of references emerge from some 9,400 hits. In the surreal real world of today, they are notably distinguished in that the authors of one set would find the content of the other to be quite meaningless, if not dangerously so:

• "algebraic" theory of correspondences: This is summarized in a standard work (E. Klein and A. C. Thompson, Theory of Correspondences, 1984) but has a long history in relation to group theory (cf Joseph Edmund Wright, Correspondences and the Theory of Groups. Transactions of the American Mathematical Society, 7, 3 1906, pp. 391-400). It is curious that the challenge of algebraic correspondences is associated in category theory with what is termed the theory of motives. A commonly applied technique in mathematics is to study objects carrying a particular structure by introducing a category whose morphisms preserve this structure. When two given objects are isomorphic, a "particularly nice" representative may be selected in each isomorphism class. A current problem in this approach is that of having 'enough' morphisms. An as yet unfulfilled hope is that development of the category of motives would lead to a universal Weil cohomology.
  
  
• "symbolist" theory of correspondences: As indicated by J E Cirlot (A Dictionary of Symbols, Dover Publications, 2002), this is founded on the assumption that: "all cosmic phenomena are limited and serial and that they appear as scales or series on separate planes; but this condition is neither chaotic nor neutral, for the components of one series are linked with those of another in their essence and in their ultimate significance. It is possible to marshal correspondences by forcing the components of any given scale or scales into a common numerical pattern: for example, it is not difficult to adapt the colour-scale from seven to eight colours, should one wish to equate it with the scale of temperaments laid down by modern character-study..."

For science in general, and mathematics in particular, progress in knowledge -- of which the algebraic variant is a generic feature -- has involved the progressive construction of a model for understanding the world that specifically disproved the validity of the premises of previous eras and notably the symbolist theory of correspondences central to those worldviews.

Classical Greece, the Gnostics and the Kabbalists are recognized as having founded much of their philosophy on the symbolist theory of correspondences -- dating back to the Egyptian "Emerald Tablet" of Hermes Trismegistus, a form of Rosetta stone in its own right. It was highly regarded by Renaissance alchemists, and a significant influence on the thinking of Isaac Newton that science has been slow to acknowledge. The "Isaac Newton" universally hailed as an exemplar of the scientific method -- notably his insights from a falling apple -- is in fact a cherry-picked version of the real Newton of larger, richer and more integrative perspective. He in fact made a translation of the Tablet which begins:

1. Tis true without lying, certain and most true.
2. That which is below is like that which is above and that which is above is like that which is below to do the miracles of one only thing.
3. And as all things have been and arose from one by the meditation of one: so all things have their birth from this one thing by adaptation.

With regard to the symbolist variant, Nathalie Wourm (The Smell of God: scent trails from Ficino to Baudelaire. 2003) notes:

This is one aspect of a theory which runs through much of European history from the Renaissance onwards, with fluctuating intensity and with fundamental variations. It has been referred to, principally, as the theory of signatures, the theory of universal analogy, and the theory of correspondences, and is originally derived from Plato's philosophy of Ideas. The most common thread of the doctrine is that there are correspondences between the material and the spiritual worlds and that the material world can therefore be read like a book, revealing the secrets of the spiritual world.

Curiously the theory of signatures, as with that of correspondences, has both a symbolist variant and a mathematical variant:

• "symbolist" theory of signatures: According to this theory (or "doctrine of signatures"), dating notably to Paracelsus, the therapeutical powers of plants were suggested. by the similarity of configuration of the affected human part. Similarity is understood as the outside manifestation of hidden relationships. Plants (as well as animals and minerals and perhaps even phenomena) were understood to have clues or "signatures" in their shapes, forms and actions indicate their purpose (The Doctrine of Signatures (Doctorine of Correspondences): the union of form and function, 1998).According to Jakob Böhme (De signatura rerum): 'Because whatever lies inside, works steadily towards its revelation; and that is the language of nature, by virtue of which every thing speaks from within its qualities, always revealing itself on its own.' Prior to the development of chemistry, it explained the active principles of medicines. The doctrine remains fundamental to homeopathy. In a discussion of similarity and similitudes, Michel Foucault (The Order of Things: an arthaeology of the human Sciences, 1970)t cites Paracelsus:

It is not God's will that what he creates for man's benefit and what he has given us should remain hidden . . . And even though he has hidden certain things, he has allowed nothing to remain without exterior and visible signs in the form of special marks -just as a man who has buried a hoard of treasure marks the spot that he may find it again (p. 25).

Drawing on the semiotic work of Peirce, Foucault, and Kristeva, Stephen H. Daniel (Philosophy of Jonathan Edwards: a study in divine semiotics, 1994) shows how the Renaissance theory of signatures provides Edwards and his contemporaries with a powerful alternative to the ideas of Descartes and Locke. The Stoic-Renaissance treatment of signs is presented as an alternative to the modern dismissal of the language of nature. The signature model could then be used in the treatment of theological themes such as creation, trinity, original sin, freedom, moral agency, and the knowledge of beauty.


• "algebraic" theory of signatures: This is especially associated with homology, a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group. Although there are many theories of cohomology, cohomology can be viewed as a method of assigning algebraic invariants to a topological space that has a more refined algebraic structure than does homology. [see also below]

A similar pattern obtains with respect to the notion of a "theory of equivalences":

• "symbolist" theory of equivalences: This is most recently associated with Eric Voegelin's theory of equivalences between various sets of historical symbolisations. Two symbolisms are there considered equivalent, despite differences of individual form, if they refer recognizably to the same structures in reality. For Voegelin, the basis of a comparative study of symbols becomes a search for the constants of experience. What is permanent in the history of mankind is not the symbols but man himself in search of his humanity and its order. "For a comparative study, if it goes beyond registering the symbols as phenomena and penetrates to the constants of engendering experience, can be conducted only by means of symbols which in their turn are engendered by the constants of which the comparative study is in search. The study of symbols is a reflective inquiry concerning the search for the truth of existential order" (Eric Voegelin, Equivalences of Experience and Symbolization in History, in: Ellis Sandoz (editor), The Collected Works of Eric Voegelin, 1990, 12, pp. 115-116)
  
  In considering the legacy of symbolism in art Charles Harrison and Paul Wood (Art in Theory 1900-1990: an anthology of changing ideas, 1996) translate a commentary by Maurice Denis (From Gauguin and van Gogh to Neo-Classicism, 1909):

  We have substituted for the idea of "nature viewed through a temperament" [Zola], the theory of equivalences or of the symbol. We affirm that the emotions or states of the soul provoked by some spectacle, create in the artistic imagination signs or plastic equivalents capable of reproducing these emotionsor states of the soul without the need to create a copy of the initial spectacle; that each state of our sensibility must correspond to an objective harmony capable of being thus translated.

  The linguist Roman Jakobson articulated a principle of equivalence through which poetry could be distinguished as "the projection of the principle of equivalence from the axis of selection to the axis of combination". This is held to imply that poetry successfully combines and integrates form and function, that poetry turns the poetry of grammar into the grammar of poetry, so to speak. Jakokbson notes (Two Aspects of Language and Two Types of Disturbances, 1956)

  Similarity in meaning connects the symbols of a metalanguage with the symbols of the language referred to. Similarity connects a metaphorical term with the term for which it is substituted. Consequently, when constructing a metalanguage to interpret tropes, the researcher possesses more homogeneous means to handle metaphor, whereas metonymy, based on a different principle, easily defies interpretation..... Since poetry is focused upon the sign, and pragmatical prose primarily upon the referent, tropes and figures were studied mainly as poetic devices. The principle of similarity underlies poetry; the metrical parallelism of lines, or the phonic equivalence of rhyming words prompts the question of semantic similarity and contrast.

• "algebraic" theory of equivalences: This is recognized in relation to process algebra, notably for regular systems. Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. They are named after Japanese mathematician Kiiti Morita who defined equivalence and a similar notion of duality in 1958. Such a theory is closely associated with group cohomology -- which has a formal definition of "splendid equivalences". In mathematics an equivalence relation is a binary relation between two elements of a set which groups them together as being "equivalent" in some way. Robin Milner (Communicating and Mobile Systems: the pi calculus, 1999) has developed a calculus for analysis of the movement of a piece of data inside a computer program -- treated exactly the same as the transfer of a message -- or indeed an entire computer program-- across the internet. Its most prominent feature is the notion of a name and is dependent on the concept of behavioural (or observational) equivalence, and the use of a new theory of types to classify patterns of interactive behaviour. Equivalence is clearly of importance in data mining in response to user questions.

But with regard to the symbolist theory of correspondences, Wourm's elaborates as follows:

The journey through minds of the theory of correspondences has obscure beginnings, but is a consequence of the Platonic hierarchy of body and soul, of a sensible world and an ideal world, and of the principle of the inherence of the non-corporeal in the corporeal. The theory relates to a quest for the spiritual meaning which is contained in each sensible object, positing that there exists a conduit between the divine and the earthly. Deriving partly from Aristotle and Plotemy's cosmology, the foundations of the theory of correspondences appear in the works of Marsilio Ficino, the fifteenth century Florentine who began the modern tradition of Neoplatonism. Ficino elaborates the idea of a world soul inherent in the cosmos, and of a system of analogies and influences between the celestial, the natural and the human worlds, which is accessible to our understanding.

It is fortunate for world culture that the rise of science, enabling the emergence of sophisticated search engines, has not resulted in the deletion of any reference to ways of knowing which science believes it has superceded. Arguably the information sciences transcend in objectivity the sciences whose production they document! Indeed, as argued by Susantha Goonatilake (Toward a Global Science: mining civilizational knowledge, 1999), it is very probable that "deprecated" metaphors natural to other cultures, such as those of the East, may in future drive and condition the formulation of theories fundamental to the sciences. Presumably such "mining" might be extended (by the East?) to "deprecated" metaphors of the West as well.

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